Convergence rates for the vanishing viscosity approximation of Hamilton-Jacobi equations: the convex case

TL;DR

This paper establishes improved convergence rates in the supremum norm for the vanishing viscosity approximation of convex Hamilton-Jacobi equations, depending on initial data regularity and Hamiltonian convexity.

Contribution

It provides new convergence rate estimates in sup-norm for convex Hamilton-Jacobi equations, extending previous $L^p$ results without relying on stochastic control or maximum principle techniques.

Findings

Convergence rate of order $oxed{ ext{O}( ext{epsilon}^eta)}$, $eta ext{ in }(1/2,1)$.

Convergence rate of order $oxed{ ext{epsilon}| ext{log} ext{epsilon}|}$.

Rates depend on initial data regularity and Hamiltonian convexity.

Abstract

We study the speed of convergence in $L^\infty$ norm of the vanishing viscosity process for Hamilton-Jacobi equations with uniformly or strictly convex Hamiltonian terms with superquadratic behavior. Our analysis boosts previous findings on the rate of convergence for this procedure in $L^p$ norms, showing rates in sup-norm of order $\mathcal{O}(\epsilon^\beta)$, $\beta\in(1/2,1)$, or $\mathcal{O}(\epsilon|\log\epsilon|)$ with respect to the vanishing viscosity parameter $\epsilon$, depending on the regularity of the initial datum of the problem and convexity properties of the Hamiltonian. Our proofs are based on integral methods and avoid the use of techniques based on stochastic control or the maximum principle.